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Wireframe, surface, solid modeling
Solid modeling gives a complete and unambiguous definition of an object,
describing not only the shape of the boundaries but also the objects interior
and exterior regions.
Disadvantages of wireframe representations
1. Ambiguous in the way to
represent an object.
2. Not suitable for
a. Mass propertycalculations
b. Hidden surface
removal
c. Shaded images
generation
Solid Representation
Real world object satisfy specific properties causing them to be
1. Bounded limited boundary, contain interior of the solid
2. Homogeneously three-dimensional no dangling edge or faces, the
boundary is always in contact with the interior of the solid.
3. Finite finite in size and limited amount of information (area, mass and
volume determinations).
Chap-678 Modeling 137
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Formal properties of geometric modeling
1. Domain or coverage define object classes
2. Validity legal model3. Completeness complete solid with enough data for geometric calculation
performed
4. Uniqueness
Solid models representation schemes
1. CSG (Constructive Solid Geometry)
2. B-Rep (Boundary Representation)
3. Sweeping4. Spatial Enumeration
Fundamental geometric principles
1. Geometry
2. Topology
3. Geometric closure
4. Set theory and operations
5. Set membership classification
Basics of Solid Modeling Theory
The fundamental geometric principles
1. Geometry and topology
Definition: Geometry relates to the information containing shape-defining
parameters, such as the coordinates of the vertices in a polyhedral object.
Definition: Topology describes the connectivity among the various
geometric components, i.e. the relational information between the different
parts of an object.
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(x,y,z) coordinates of vertices geometry
connectivity matrix topology
2. Geometric closure Bounded, Finite, No dangling
3. Set Theory
A set is defined as any collection of objects, called elements or members.
Universal setW, containing all points in 3E space, and the null set, , no
elements.Set operations: union (), intersection (), difference ().
4. Regularized set operations Boolean operations ensure the validity of
geometric models, avoiding the creation of nonsense objects.
5. Set membership classification two sets X and S, check how various
parts of X can be assigned to S as being on its interior, exterior, or on its
boundaries. X is partitioned into subsets XinS, XonS, XoutS.
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Constructive Solid Geometry (CSG)
A CSG model assumes that physical objects can be created by combining
basic elementary shapes (primitives) through specific rules.
CSG primitives are represented by the intersection of a set of half-spaces, as
shown in Figure 12.10.
Quadric surfaces are commonly used in CSG because they represent the most
commonly used surface in mechanical design produced by the stand
operations of milling, turning, rolling. E.g. planar surfaces are obtained through
rolling and milling, cylindrical surface through turning, spherical surfaces
through cutting done with a ball-end cutting tool.
Data structure for the CSG representation is based on the binary tree
structure.
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CSG example
Disadvantages:1. The way of primitive combinations for the CSG representation is not unique.
It has been found through the use of different primitives and Boolean
operations.
2. The CSG doesnt specify quantitative values for the new solid (unevaluated
model). The new model must be checked through a boundary evaluation
routine with will supply quantitative information about its vertices, edges,
faces.
3. It shorts of intersection calculation in the form of curve/curve, curve/surface,
or surface/surface intersections.
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Boundary Representation B-rep
The B-rep is built on the idea that a physical object is enclosed by a set of
faces, which themselves belong to closed and orientable surfaces.
Geometric and topology entities
Point Vertex
Curve Edge
Surface Face
The Eular-Poincar law gives a quantitative relationship among faces, edges,
vertices, faces inner loops, bodies or through holes (genus) in solids.
The Eular-Poincar lawF-E+V-L = 2(B-G)
A loop represents a connected portion of the boundary of a face. The faces
inner loop represents the connected portion of the boundary of two faces.
Eular law is not only suit for solids with planar faces, but also for curved objects
with closed curved faces or edges.
Simplest form : F-E+V=2
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Spatial enumeration schemes
The smaller the cell is, the more accurate the model is.
Spatial enumeration schemes have the advantage of easy access to any part
of the model and the assurance of spatial uniqueness.
2D quadtree
3D octree
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Solid modeling systems
Comparison between CSG and B-rep representations.
Storage of Model Detail Level
CSG Implicit Low
B-rep Explicit High
Advantages (A) and Disadvantages (D) comparisons.
Complexity Uniqueness History of
Construction
Use in
Interactive
Environment
Local
Operations
CSG A D A D D
B-rep D A D A A
Conversion among representations.
Solid modeling systems are characterized as mostly CSG or
mostly B-rep. (Dual representation of a solid model)
Mostly CGS model creation and editing is done only in the CSG form; once
the model is created, a boundary evaluator algorithm is used to obtain a
boundary representation (store internally along with the CSG tree).
Mostly B-rep the user can create the model in either CSG or B-rep, but the
CSG representation is discarded by the system.
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NURBS can exactly represent quadric surfaces, so internal operations in the
modeler, such as the calculation of surface/surface intersection are
accomplished with a single algorithm. (minimize the amount of geometric
software required in the modeler)
Feature modeling
Features can link CAD and CAM in an efficient way.
The feature modeler contains not only a geometric and topological structure
but also support geometric characteristics of a part. (shapes of holes, cutouts,
slots, chamfers, ribs, etc.)
Three fundamental approaches to feature modeling
1. Human-assisted feature recognition
E.g. Tolerance or surface of model are created and stored in the database and
later used by process planning systems.
2. Automatic feature recognition
Find and extract form features the correspond to some predefined geometric
pattern. (very difficult)
3. Design by feature
Applications of Solid Modeling to Engineering
Solid modeling is commonly used in engineering to aid visual analysis of a
design idea, mass property calculations, and static interference analysis.
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A simple Solid Modeling System (sample)
Four steps for developing the solid modeling system
1. CSG representation for user input (create quadric primitives)
2. Conversion to a faceted representation
3. Intersection calculation Boolean operations
4. Rendering
Recommended